# Professor Philip Broadbridge

F.AustMS

Professor

College of Science, Health and Engineering

School of Engineering and Mathematical Sciences

Department of Mathematics and Statistics

Melbourne (Bundoora)

### Qualifications

BSc(hons)(Adel), DipEd(Tas), PhD(Adel)

### Role

Academic

### Membership of professional associations

Past Pres ANZIAM, Fellow Aust Math Soc, Member AIP, IAMP

### Area of study

Mathematics and Statistics

### Brief profile

Phil Broadbridge rejoined La Trobe in 2009 as Head of the School of Engineering and Mathematical Sciences. From 2005 to 2009, he was Director and CEO of the Australian Mathematical Sciences Institute and in that capacity was presented with the 2008 Fast Thinking/ Open Universities National Innovation Award in the category of science innovation. He was previously a professor of applied mathematics for 14 years, including a total of 8 years as department chair at the University of Delaware and at the University of Wollongong, where he was also Deputy Chair of Academic Senate. Having worked also as a secondary teacher and as a CSIRO researcher, he has strong interests in education and in industry outreach. He is particularly interested in the links between engineering and mathematics, and was part of the original writing team that secured HEIF funding for SMART Infrastructure at the University of Wollongong.

His PhD was in mathematical physics (University of Adelaide). He has an unusually broad range of research interests, including fundamental physics, applied nonlinear partial differential equations, hydrology, heat and mass transport and population genetics. He has published two books and over 100 refereed papers, appearing in 30 different scientific journals. He is a member of the editorial boards of two book series and three journals, including Proceedings of the Royal Society of London. He has been Editor-in-Chief of the US-based journal, *Mathematical and Computer Modelling*, as well President of Australia New Zealand Industrial and Applied Mathematics, the main professional association of applied mathematicians in Australia and New Zealand.

### Research interests

#### Hydraulics and Water Resources

- Please contact me to discuss a topic.#### Integrable systems

- Please contact me to discuss a topic.#### Mathematical foundations of physics

- Please contact me to discuss a topic.### Teaching units

current: MAT2VCA Vector Calculus and Laplace transforms. MAT1CDE Calculus and Differential Equations. MAT4PDE Nonlinear Partial Differential Equations.

past: MAT3DQM Dynamical Systems and Quantum Mechanics, 2nd year Dynamics

At other places: 1st year applied maths modelling, 3rd year industrial modelling, 3rd year continuum mechanics, 3rd year boundary value problems

### Consulting

experience with BHP Research, APRA, Cadburys, Southcorp Wines.

### Recent publications

N. M. Ivanova and P. Broadbridge, “Solutions and reductions for radiative energy transport in laser-heated plasma”, J. Math. Phys. **56**, 011503 (2015); http://dx.doi.org/10.1063/1.4905646

E. Tonkes and P. Broadbridge, “A parsimonious diffusion equation for electricity demand”, ANZIAM Jnl 54 (CTAC 2012), pp C747-C767 (2014).

P. Broadbridge, C. Chanu and W. Miller Jr., “Solutions of Helmholtz and Schrodinger equations with a side condition and nonregular separation of variables”, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) Vol 8, article 089 (31 pages), special issue celebrating Peter Olver’s 60th birthday (2012).

N. Tehseen and P. Broadbridge, “Classification of Fourth Order Diffusion Equations with Increasing Entropy”, Entropy 2012, 14(7), 1127-1139; doi:10.3390/e14071127

D. Triadis and P. Broadbridge, "The Green-Ampt limit with reference to infiltration coefficients", Water Resour Res. , special issue in honour of J.-Y. Parlange and W. Brutsaert, 48 (7), DOI: 10.1029/2011WR011747 (2012).

P. Broadbridge and J. M. Goard, “When central finite differencing gives complex values for a real solution !”, Complex Variables and Elliptic Equations, 57 (2-4), 455-467 (2012).

P. Broadbridge and P. J. Vassiliou, “The Role of Symmetry and Separation in Surface Evolution and Curve Shortening”, invited paper for special issue in honour of Willard Miller Jr., P. Olver, P. Winternitz and E. Kalnins (guest editors), Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 7, 052, 19 pages (2011).

E. Haque and P. Broadbridge, “Exact Solution of a Boundary Value Problem Describing the Uniform Cylindrical or Spherical Piston Motion”, Applied Math Modelling, 35, 3434-3442 (2011).

E. Haque, P. Broadbridge and P. L. Sachdev, “Expansion of High Pressure Gas into Air- A More Realistic Blast Wave Model”, Math. Comput. Modelling, 50, 1606-1621 (2009).

P. Broadbridge and J. M. Goard, “Grain Boundary Evolution with Time Dependent Material Properties”, J. Eng. Math., 66(1-3), 87-102 (2010).

D. Triadis and P. Broadbridge, “Analytical model of infiltration under constant-concentration boundary conditions”, Water Resour Res., 46(3), W03526, doi:10.1029/2009WR008181 (2010).

P. Broadbridge, “Entropy behaviour of evolution PDEs”, PAMM (Proceedings of Applied Mathematics and Mechanics), 7(1), 2040027-2040028 (2008).

S. Henderson and P. Broadbridge, “Engineering Mathematics Education in Australia”, MSOR Connections (UK), 9(1), 12-17, 2009.

P. Broadbridge, D. Triadis and J.M. Hill, “Infiltration from supply at constant concentration: An integrable model”, invited article for special edition on soil-water transport, J. Eng. Math, 64(2), 193-206 (2009).

P. Broadbridge, “Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form”, Entropy, 8, 295-311 (2008). http://www.mdpi.org/entropy/papers/e10030365.pdf

B. H. Bradshaw-Hajek, P. Broadbridge and G. H. Williams, “Evolving gene frequencies in a population with three possible alleles at a locus”, Mathematical and Computer Modelling, 47, 210-217 (2008).

B. H. Bradshaw-Hajek, M.P. Edwards, P. Broadbridge and , G.H. Williams, “Nonclassical symmetry solutions for reaction-diffusion equations with explicit spatial dependence”, Nonlinear Analysis: Theory, Methods & Applics. 67, 2541-2552 (2007).

R. J. Moitsheki, P. Broadbridge and M.P. Edwards, “Group invariant solutions for two dimensional solute transport under realistic water flows”, Quaestiones Mathematicae, 29(1), 73-83 (2006).

P. Broadbridge and P. Zulkowski, “Dark Energy States from Quantization of Boson Fields in a Universe with Unstable Modes”, Rep. On Math. Phys., 57 (1), 27-40 (2006).

R. J. Moitsheki, P. Broadbridge and M. P. Edwards, "Solutions for Transient Solute Transport in Unsaturated Soils During Steady Evaporation from a Water Table", Transp. In Porous Media 61(1), 109-125 (2005).

R. J. Moitsheki, P. Broadbridge and M.P. Edwards, “Systematic construction of hidden nonlocal symmetries for inhomogeneous nonlinear diffusion equations”, J. Phys. A 37(34), 8279-8286 (2004).

J. M. Goard, P. Broadbridge and G. Raina, "Tractable forms of the Bond Pricing Equation", Math. Comput. Modelling.. 40(1-2), 151-172 (2004).

B. H. Bradshaw-Hajek and P. Broadbridge, "A robust cubic reaction-diffusion system for gene propagation.", Mathematical and Computer Modelling, 39(9-10), 1151-1163 (2004).

J. M. Goard and P. Broadbridge, "Exact solution of a degenerate fully nonlinear diffusion equation", Journal of Appl. Math. and Physics (Z.A.M.P.) 55(3), 534-538 (2004) .

P. Broadbridge, B. Bradshaw, G. Fulford, and G. K. Aldis, "Huxley and Fisher Equations for Gene Propagation: An Exact Solution ", The ANZIAM Journal 44 (1) 11-20 (2002).

J. M. Stewart, P. Broadbridge and J. M. Goard, " Symmetry Analysis and Numerical Modelling of Invasion by Malignant Tumour Tissue", Nonlinear Dynamics, 28 (2), 175-193 (2002).

Lance Miller and P. Broadbridge, "Exact Integration of Reduced Fisher's Equation, Reduced Blasius Equation and the Lorenz Model", J. Math. Anal. Applic., 251, 65-83 (2000).

P. Tritscher, W. W. Read, P. Broadbridge and J.H. Knight,"Steady Saturated-Unsaturated Flow in Irregular Porous Domains", Math. Comp. Modelling, 34, Issue 1/2, 177-194 (2001).

P. Broadbridge, J.M. Hill and J.M. Goard, "Symmetry Reductions of Equations for Solute Transport in Soil", Nonlinear Dynamics, 22, 15-27 (2000).

Ali El-Feki, P. Broadbridge and G.W. Walter, "Potential Transients for an Electrochemical Corrosion Reaction under Constant Current Conditions" , Mathl. & Computer Modelling, 30 (11-12), 111-131 (1999).

### Resume

### Older publications

P. Tritscher, W.W. Read and P. Broadbridge, "Specific Yield for a Two-Dimensional Flow", Water Resour. Res., **36**, 1393-1402 (2000).

J. M. Goard and P. Broadbridge, "Solutions to Nonlinear PDES from Symmetry-Enhancing and Symmetry-Preserving Constraints", J. Math. Anal. Applic., 238, 369-384 (1999).

P. Broadbridge, M.M. Lavrentiev Jr. and G. H. Williams, "Nonlinear Heat Conduction through an Externally Heated Radiant Plasma: Background Analysis for a Numerical Study", J. Math. Anal. Applic. , 238, 353-368 (1999).

P. Broadbridge and D. J. Arrigo, "All solutions of standard symmetric linear partial differential equations have classical lie symmetry", J. Math. Anal. Applic. , 234, 109-122 (1999).

P. Broadbridge, " The forced Burgers equation, plant roots and Schrödinger's eigenfunctions", J. Eng. Math., 36 (1), 25-39 (1999).

P. Broadbridge, G.R. Fulford, N. D. Fowkes, D.Y.C. Chan and C. Lassig, "Bubbles in Wet Gummed Wine Labels" , SIAM Review, 41 (2), 363-372 (1999).

J. M. Stewart and P. Broadbridge, "Calculation of humidity during evaporation from soil", Adv. Water Resour. , 22, 495-505 (1999).

A. Ali El-Feki, P. Broadbridge and G.W. Walter, "Exact solutions of the potentiostatic current transients for a corrosion reaction under mixed charge transfer and diffusion control", Math. Comp. Modelling, 29 (3), 27-421(1999).

P. Tritscher, W. W. Read, P. Broadbridge and J. M. Stewart, "Steady Infiltration in Sloping Porous Domains: The Onset of Saturation", Transp. Porous Media, 31 (1), 1-17 (1998).

D.J. Arrigo, P. Broadbridge, P. Tritscher and Y. Karciga, "The Depth of a Steep Evaporating Grain Boundary Groove: Application of Comparison Theorems", Math. Comp. Modelling, 25 (10), 1-8 (1997).

P. Broadbridge and J. M. Goard, "Steady Unsaturated Flow in Two-Dimensional Scale-Heterogeneous Porous Media", Math. Comp. Modelling, 26 (3), 45-54 (1997).

J.M. Goard and P. Broadbridge, "Nonlinear Superposition Principles Obtained by Lie Symmetry Methods", J. Math. Anal. Appl., 214, 633-657 (1997).

M.Lavrentiev Jr., V. Belov and P. Broadbridge, "Boundary Value Problems for Strongly Degenerate Parabolic Equations", Commun. Partial Differential Equations, 22, 17-38 (1997).

P. Broadbridge, J. M. Goard and M. Lavrentiev Jr., "Degenerate Nonlinear Diffusion with an Initially Sharp Front", Stud. Appl. Math., 99, 377-391 (1997).

J. M. Goard, P. Broadbridge, and D.J. Arrigo, "The Integrable Nonlinear Diffusion Equation U_t = [f(U) / U_x]_x and its Relatives" , J. of Appl. Maths. & Physics (ZAMP), 47, 926-942 (1996).

P. Broadbridge, M. P. Edwards and J.E. Kearton, "Closed-form Solutions for Unsaturated Flow under Variable Flux Boundary Conditions", Adv. Water Resour., 19, 207-213 (1996).

D.J.Arrigo, J.M.Goard and P. Broadbridge, "Nonclassical Solutions are Non-Existent for the Heat Equation and Rare for Nonlinear Diffusion", J. Math. Anal. Applic.,202, 259 - 279 (1996).

W.W.Read and P.Broadbridge, "Series Solutions for Steady Unsaturated Flow in Irregular Porous Domains", Transp. in Porous Media, 22, 195-214 (1996).

P.Broadbridge and B. M. Pincombe, " The Stefan Solidification Problem with Nonmonotonic Nonlinear Heat Diffusivity" , Math. & Comp. Modelling, 23 (10), 87-98 (1996).

M.P. Edwards and P. Broadbridge, "Exceptional Symmetry Reductions of Burgers' Equation in Two and Three Spatial Dimensions", J. of Appl. Maths. & Physics (ZAMP),

46, 595-622 (1995).

P. Tritscher and P. Broadbridge, "Grain Boundary Grooving by Surface Diffusion : an Analytic Nonlinear Model for a Symmetric Groove", Proc. Roy. Soc. A, 450, 569-587 (1995).

J.M. Goard and P. Broadbridge, "Nonclassical Symmetry Analysis of Nonlinear Reaction- Diffusion Equations in Two Spatial Dimensions", Nonlin. Analysis : Theory, Methods & Applications, 26, 735-754 (1996).

M.P. Edwards and P. Broadbridge,"Exact Transient Solutions to Nonlinear Diffusion-Convection Equations in Higher Dimensions", J. Phys. A. : Math. Gen. , 27, 5455-5465 (1994).

P. Broadbridge and P. Tritscher, " An integrable fourth order nonlinear evolution equation applied to thermal grooving of metal surfaces", I.M.A. J. Appl. Math., 53, 249-265 (1994).

P. Tritscher and P. Broadbridge, "A Similarity Solution of a Multiphase Stefan Problem Incorporating General Nonlinear Heat Conduction", Int. J. Heat Mass Transfer., 37, 2113-2121 (1994).

D. J. Arrigo, J.M. Hill and P. Broadbridge, "Nonclassical Symmetry Reductions of the Linear Diffusion Equation with a Nonlinear Source ", I.M.A. J. Appl. Math, 52, 1-24 (1994).

D. J. Arrigo, J. M. Hill and P. Broadbridge, "Nonclassical Symmetry Solutions and the Methods of Bluman-Cole and Clarkson-Kruskal", J. Math. Phys. ,34, 4692-4703 (1993).

P. Broadbridge, P. Tritscher and A. Avagliano, "Free Boundary Problems with Nonlinear Diffusion", Mathl. Comp. Modelling, 18, 15-34 (1993).

P. Broadbridge and C. Rogers, "On a Nonlinear Reaction-Diffusion Boundary-Value Problem: Application of a Lie-Bäcklund Symmetry", J. Austral. Math. Soc. (b) , 34, 318-332 (1993).

P.Broadbridge, "Reply to Barry et al ", Water Resour. Res. , 28, 1739-1740 (1992).

P. Broadbridge, R. Srivastava and T.-C. Yeh, "Burgers' Equation and Layered Media: Exact Solutions and Applications to Soil-Water Flow", Mathl. Comp. Modelling , 16, 163-169 (1992).

C. Rogers and P. Broadbridge, "Sedimentation in a Bounded Column" , Internat. J. Nonlin. Mech. , 27, 661-667 (1992).

P. Broadbridge and P.J. Banks, "Exact Nonlinear Solution for Constant-Rate Expression from Material of Finite Thickness", J. Austral. Math. Soc. (B), 33, 430-450 (1992).

A. Warrick and P. Broadbridge, "Sorptivity and Macroscopic Capillary Length Relationships", Water Resour. Res. 28, 427-431 (1992).

A.W. Warrick, P. Broadbridge and D.O. Lomen, "Approximations for Diffusion from a Disk Source", Applied Math. Modelling, 16, 155-161 (1992).

P. Broadbridge and S.E. Godfrey, "Integrable Heterogeneous Nonlinear Schrödinger Equations with Dielectric Loss: Lie-Bäcklund Symmetries", J. Math. Phys. 32, 8-18 (1991).

P. Broadbridge, "Solution of a Nonlinear Absorption Model of Mixed Saturated-Unsaturated Flow", Water Resour. Res., 26, 2435-2443 (1990).

P. Broadbridge, "Infiltration in Saturated Swelling Soils and Slurries; Exact Solution for Constant Supply Rate", Soil Sci. 149, 13-22 (1990).

P. Broadbridge and C. Rogers, "Exact Solutions for Vertical Drainage and Redistribution in Soils", J. Eng. Math. 24, 25-43 (1990).

P. Broadbridge, "Exact Solvability of the Mullins Nonlinear Diffusion Model of Groove Development", J. Math. Phys. 30, 1648-51 (1989).

I. White and P. Broadbridge, "Reply to Comment by Smith and Parlange on Constant Rate Rainfall Infiltration", Water Resour. Res. 25, 1054-59 (1989).

P. Broadbridge and I. White, "Reply to Comment by Sander et al on Constant Rate Rainfall Infiltration", Water Resour. Res. 24, 2109-10 (1988).

P. Broadbridge, J.H. Knight and C. Rogers, "Constant Rate Rainfall Infiltration in a Bounded Profile; Exact Solutions of a Nonlinear Model", Soil Sci. Soc. Am. J. 52, 1526-33 (1988).

P. Broadbridge, "Integrable Forms of the One-Dimensional Flow Equation for Unsaturated Heterogeneous Porous Media", J. Math. Phys. 29, 622-27 (1988).

C. Rogers and P. Broadbridge, "On a nonlinear moving boundary problem with heterogeneity: application of a Bäcklund transformation", J. of Appl. Maths. & Physics (ZAMP) 39, 122-128 (1988).

P. Broadbridge and I. White, "Constant Rate Rainfall Infiltration: A Versatile Non-linear Model. 1. Analytic Solution", Water Resour. Res. 24, 145-54 (1988).

I. White and P. Broadbridge, "Constant Rate Rainfall Infiltration: A Versatile Non-linear Model. 2. Applications of Solutions", Water Resour. Res. 24, 155-162 (1988).

P. Broadbridge and I. White, "Time to Ponding: Comparison of Analytic, Quasi-analytic and Approximate Predictions", Water Resour. Res. 23, 2302-10 (1987).

P. Broadbridge, "Integrable Flow Equations that Incorporate Spatial Heterogeneity", Transp. Porous Media 2, 129-144 (1987).

P. Broadbridge, "Nonintegrability of Nonlinear Diffusion-Convection Equations in Two Spatial Dimensions", J. Phys. A: Math. Gen. 19, 1245-57 (1986).

P. Broadbridge, "Quantization of Gyroscopically Stable Systems", Annals Phys. (N.Y.) 168, 273-83 (1986).

P. Broadbridge, "Algebraic Quantization with Indefinite Metric", J. Phys. A: Math. Gen. 16, 3271-90 (1983).

P. Broadbridge, "Existence Theorems for Segal Quantization via Spectral Theory in Krein Space", J. Austral. Math. Soc. (b) 24, 439-60 (1983).

P. Broadbridge, "Quantization, Non-Locality and Lie-Admissible Formulations", Hadronic J. 5, 1842-58 (1982).

P. Broadbridge and C.A. Hurst, "Fermi-Dirac Quantization of Linear Systems", Annals Phys. (N.Y.) 137, 86-103 (1981).

P. Broadbridge and C.A. Hurst, "Canonical Forms for Quadratic Hamiltonians", Physica A108, 39-62 (1981).

P. Broadbridge, "Problems in the Quantization of Quadratic Hamiltonians", Hadronic J. 4, 899-948 (1981).

P. Broadbridge and C.A. Hurst, "Existence of a complex Structure for Quadratic Hamiltonians?", Annals Phys. (N.Y.) 131, 104-17 (1981).

P. Broadbridge, "Normal Forms for Classical and Boson Systems", Physica A99, 494-512 (1979).

P. Broadbridge, "Photons as Parafermion Composites - Limitations", Reports on Math. Phys. 14, 179-83 (1978).

### Research projects

1. Soil-water-solute flow modelling. Information theory and robust water distributions.

2. The role of total population drift in the spread of a new gene.

3. Exact solutions for solute transport via incomplete Staeckel matrices.

4. Exact solutions for nonlinear equations of Kuramoto-Sivashinsky type: a balance between forward and backward diffusion at different orders.

5. Turbulent fluid statistics from approximating systems of ODEs.

6. Unstable modes in quantum mechanics.

6. Discrete analogues of classically integrable nonlinear partial differential equations.